Meeting your requirements:
✓ Usable for desktop and server
It has a GUI for desktop use: X-Windows, with a choice of Gnome or KDE Desktop environments; or Cinnamon, Mate or Xfce.
It is designed as an enterprise operating system with high performance on the server. The web hosting company I use, uses it, as do many others, on their servers.
In my experience, using it on both, smoothes the path between local dev and server deployment.
✓ Open source
CentOS is open-source, gratis and libre: you can download CentOS here, and all the source is here
✓ Long-Term Support releases (at least 3 years' support)
CentOS Linux has a ten-year support cycle. The most recent CentOS, 7, will be supported until 30 June 2024
✓ Focused on quality
? bugfixes include tests to ensure bugs don't come again
✓ Some kind of Continous Integration to be sure only packages which pass all automated tests get released.
Its raison d'être is long-term stability for the enterprise, so offers you the robustness and long life that you are asking for.
It does daily continuous integration testing. The test suite itself lives on github
Linux veterans will have known of this in the olden days as Red Hat. Red Hat Enterprise Linux became a commercial product. CentOS is the free (gratis & libre) version of Red Hat Enterprise Linux.
My personal experience
(disclosure: I have no connection to CentOS except as a very satisfied user)
I've just switched to it after years of wrestling with kubuntu. It meets my needs as a development machine that I can build and test stuff on before deploying to a server. All the packages I'd expect to find, are available. I recommend adding the Extra Packages for Enterprise Linux (EPEL) repository to your list of sources (and that's easy to do from the command-line:
sudo yum install epel-release)
This one is pretty much inevitable when you're looking for long-term stability: if I want the latest version of a package (say Python 3.5, PHP 7, gcc 5.2 at time of writing), then I do have to get the source package and build it myself. If I can be satisfied with an earlier version, then a yum package is available from the CentOS repositories.
The support community is big, but it's not as huge as for ubuntu. It does contain a lot of professional web hosts, and fewer hobbyists. There is a Centos tag on the Unix Stackexchange, whereas there's a whole Stack dedicated to the various *ubuntus.